Integration is sometimes guesswork. In the case of the integral of e^3x, you need to find a function F(x) which when differentiated produces e^3x. So if I guess F(x) = e^3x, then F'(x) = 3e^3x. So it appears than F'(x) is 3 times more than e^3x. So I will guess F(x) = e^3x / 3, then F'(x) = 3e^3x / 3 = e^3x. So the integral of e^3x is e^3x / 3.